Physical Dynamics and Constraints › Conservation Laws and Symmetries

Conservation laws are powerful structural priors. For systems described by a variational principle, Noether's theorem links differentiable continuous symmetries of the action, including invariance up to an appropriate boundary term, to quantities conserved along solutions.

Noether's theorem turns conservation from a checklist into a diagnostic. If a model predicts spontaneous acceleration for an isolated, force-free system, it may have encoded position dependence with no physical source or omitted an interaction with the environment. For an open subsystem, however, external fields and unmodeled forces can change momentum without implying a broken symmetry of the full system. We will use energy as one evaluation tool in the worked example. Drift can come from the learned vector field, the integrator, finite step size, external forcing, or evaluating a physical energy different from the scalar the model conserves. Noether identifies the symmetry–conservation relationship; diagnosis still requires separating those sources.

There is a subtlety that trips up even careful modelers. Energy conservation is a property of an autonomous conservative system's exact flow, while numerical integration discretizes time. Explicit Euler generally does not preserve the Hamiltonian and produces secular energy growth for the harmonic oscillator and many pendulum trajectories, although the sign of a one-step energy error is not universally positive. Symplectic integrators such as semi-implicit Euler and velocity Verlet preserve the symplectic form. For sufficiently smooth autonomous Hamiltonians, small fixed steps, and bounded trajectories, backward-error analysis explains why their numerical energy often stays close to a nearby shadow Hamiltonian for very long times. Symplectic methods are therefore widely used for long-term Hamiltonian simulation, while implicit, adaptive, and other nonsymplectic schemes remain appropriate in different regimes.

The lesson for world models is architectural as well as numerical. An explicit-Euler-like update generally does not preserve Hamiltonian structure. A valid symplectic map applied to a suitable Hamiltonian with a stable step size can instead provide long-time near-conservation behavior. Update structure does not rescue a wrong vector field, and bounded energy does not imply small state error, but it removes one identifiable source of secular drift.

Symmetries can also be built into the function approximator. An equivariant network commutes with a symmetry group: transforming the input transforms the output in the corresponding way. For gravity-aware dynamics, coordinate rotations must transform the geometry, vector features, outputs, and gravity vector together; rotating the bodies while holding gravity fixed is a physically different problem. Momentum-conserving interaction networks, rotation-equivariant point-cloud models, and SE(3)-aware dynamics networks build on this idea. For an isolated pairwise model, equal-and-opposite learned impulses can enforce total-momentum conservation by construction, provided external forces and non-pairwise interactions are handled separately.